Axiom ax-mulass 8230

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8188. Proofs should normally use mulass 8258 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 8125	. . . 4 class ℂ
3	1, 2	wcel 2203	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2203	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2203	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1005	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 8132	. . . . 5 class ·
10	1, 4, 9	co 6050	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6050	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6050	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6050	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1398	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8258